On invariant subspaces of several variable Bergman spaces

Tohoku Mathematical Journal, 1996

We show that the Bergman kernel function, associated to pseudoconvex domains of finite type with the property that the Levi form of the boundary has at most one degenerate eigenvalue, is a standard kernel of Calderón-Zygmund type with respect to the Lebesgue measure. As an application, we show that the Bergman projection on these domains preserves some of the Lebesgue classes.

arXiv (Cornell University), 2022

Consider a bounded, strongly pseudoconvex domain D ⊂ C n with minimal smoothness (namely, the class C 2) and let b be a locally integrable function on D. We characterize boundedness (resp., compactness) in L p (D), p > 1, of the commutator [b, P ] of the Bergman projection P in terms of an appropriate bounded (resp. vanishing) mean oscillation requirement on b. We also establish the equivalence of such notion of BMO (resp., VMO) with other BMO and VMO spaces given in the literature. Our proofs use a dyadic analog of the Berezin transform and holomorphic integral representations going back (for smooth domains) to N.

Integral Equations and Operator Theory, 1999

Let II be the upper half-plane in C, consider the Bergman space A2(II), the subspace of all analytic functions from Lz(II). The complete decomposit~on of L2(II) onto Bergman and Bergman type spaces of poly-analytic and poly-antianalytic functions is obtained. The orthogonal Bergman type projections onto each of these subspaces are described. Connections with the Hardy spaces and the Szego projections are established. where Pn-l in the orthogonal projection (4.6) of LI(R+) onto the one-dimensional space Ln-1, generated by the function I,-l(y), defined in (4.2). PROOF. Follows directly from Corollaries 4.2 and 4.4.

The following text is a modified and updated version of the problem collection , which was written in 1993 but became publicly available only in 1995. It was a survey of various open problems; a general survey of the field was provided in [41, 42] in 1998, written in 1995 and 1996, respectively. Since then, a number of new developments have taken place, which in their turn have led to new questions. We feel it is time to update the problem collection.

2015

The purpose of this paper is to prove L^p-Sobolev and Hölder estimates for the Bergman projection on a class of pseudoconvex domains that admit a "good" dilation and satisfy Bell-Ligocka's Condition R. We prove that this class of domains includes the h-extendible domains, a large class of weakly pseudoconvex domains of finite type.

Journal of Functional Analysis, 1985

Indiana University Mathematics Journal, 2002

A theorem of Aleman, Richter, and Sundberg asserts that every z-invariant subspace M of the Bergman space A 2 is generated by M zM, the orthocomplement of zM within M. The purpose of this paper is investigate the extent to which that property generalizes to g-invariant subspaces for a function g ∈ H ∞. Such a function g is said to have the wandering property in A 2 if every g-invariant subspace M of A 2 is generated by M gM. In the Hardy space H 2 , every inner function has the wandering property, while every function with this property must be the composition of an inner function with a conformal mapping. In this paper it is shown that the only functions that can have the wandering property in A 2 are essentially the classical inner functions. On the other hand, a large class of inner functions for which this property fails is exhibited. The wandering property is equivalent to the cyclicity of certain reproducing kernels; thus the proofs involve the approximation of noncyclic kernels by kernels corresponding to pullback measures.

We define a set of projections on the Bergman space A 2 parameterized by an affine closed space of a Banach space. This family is defined from an affine space of a Banach space of holomorphic functions in the disk and includes the classical Forelli-Rudin projections.

2015

Additionally we prove that the conjecture is true whenever M is a nontrivial, zero based invariant subspace of L 2 a (D). Furthermore it is shown that if I = T\σ(M * ζ | M ⊥) = ∅, then l λ for fixed λ ∈ D, has a meromorphic continuation across I. We also provide examples when some extra hypothesis are imposed on l λ and we obtain information for the invariant subspaces related to them. Contents Chapter 1. Introduction 1.1. Banach Spaces of Analytic Functions 1.2. Invariant Subspaces and the Index Function 1.3. The General Theory of Reproducing Kernels 1.4. Bergman Spaces 1.5. Overview and Introduction to the Main Results Chapter 2. Reproducing Kernels and their Rank 2.1. Theory of B-kernels 2.2. Positive Operators and Reproducing Kernels 2.3. The Main Theorem Chapter 3. Properties of the Reproducing Kernels 3.1. The Cauchy Transform 3.2. Properties of the Kernel Functions Chapter 4. Applications and Further Results 4.1. Applications 4.2. Hedenmalm's Conjecture Bibliography Vita v CHAPTER 1.1. Banach Spaces of Analytic Functions Let Ω be an open connected subset of the complex plane C and let Hol(Ω) be the space of analytic functions on Ω. 1.2. Invariant Subspaces and the Index Function If T is an operator on a Banach space X, then a closed subspace M of X is called invariant for T if T M ⊂ M. The collection of all invariant subspaces of an operator T is denoted by Lat T. It forms a complete lattice with respect to intersections and closed spans. From now on suppose that Ω is a bounded, non-empty region in the complex plane and that B is a Banach space of analytic functions on Ω satisfying axioms (1) − (4) of the previous section. By an invariant subspace we will always mean an invariant subspace for (M ζ , B), unless it is stated otherwise. For a subset S of B write [S] for the smallest invariant subspace which contains all of S. For a single Hilbert space we set M ⊖ ζM = M ∩ (ζM) ⊥. Note that M ⊖ ζM is Hilbert space isomorphic to M/ζM and hence ind M = dim(M ⊖ ζM). Theorem 1.2.3. Suppose f ∈ B, f = 0. Then ind [f ] = 1. Suppose that A = {α k } k∈N ∈ Ω is a B-zero sequence; that is the sequence of zeros, repeated according to multiplicity, of some nonidentically vanishing function in B. see [1, Theorem 3.2], that if M ∈ Lat (M ζ , L p a (D, (1 − |z|) α dA)) contains a nonzero function which is locally Nevanlinna, then ind M = 1. In addition to the above and in relation to Beurling's theorem, Aleman, Richter and Sundberg showed in [3, Theorem 3.5] that if M ∈ Lat(M ζ , L 2 a), then M = [M ⊖ ζM]. ∞ j,i=0k (j, i)z j λ i (λ, z) ∈ D × D. Then k is a reproducing kernel on D×D if and only if the infinite matrix {k(j, i)} ∞ j,i=0 is positive definite. 1.4. Bergman Spaces The weighted Bergman spaces L p a (Ω, wµ A) were considered in Example 2 of the first section. For 0 < p < +∞ and −1 < α < +∞ the weighted Bergman space L p a,α of the disc D, is the space of analytic functions in L p (D, dA α) where dA α (z) = (α + 1)(1 − |z| 2) α dA(z). Indeed, if G M i are unit vectors in M i ⊖ ζM i , i = 1, 2, then M 1

The Journal of Geometric Analysis, 2019

We study the restriction operator from the Bergman space of a domain in C n to the Bergman space of a non-empty open subset of the domain. We relate the restriction operator to the Toeplitz operator on the Bergman space of the domain whose symbol is the characteristic function of the subset. Using the biholomorphic invariance of the spectrum of the associated Toeplitz operator, we study the restriction operator from the Bergman space of the unit disc to the Bergman space of subdomains with large symmetry groups, such as horodiscs and subdomains bounded by hypercycles. Furthermore, we prove a sharp estimate of the norm of the restriction operator in case the domain and the subdomain are balls. We also study various operator theoretic properties of the restriction operator such as compactness and essential norm estimates.